Uniform error bound of an exponential wave integrator for the long-time dynamics of the nonlinear Schrödinger equation with wave operator

Abstract

We establish the uniform error bound of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Schrödinger equation with wave operator (NLSW), in which the strength of the nonlinearity is characterized by ${\epsilon }^{2p}$ with $\epsilon \in (0,1]$ a dimensionless parameter and $p\in {\mathbb{N}}^{+}$. When $0<\epsilon \ll 1$, the long-time dynamics of the problem is equivalent to that of the NLSW with $O(1)$-nonlinearity and $O(\epsilon )$-initial data. The NLSW is numerically solved by the EWI-FP method which combines an exponential wave integrator for temporal discretization with the Fourier pseudospectral method in space. We rigorously establish the uniform $H^1$-error bound of the EWI-FP method at $O(h^{m-1}+{\epsilon }^{2p-{\hat{I}}^2}\ddot{I}{„}^2)$ up to the time at $O(1/{\epsilon }^2)$ with $0\le {\hat{I}}^2\le 2p$, the mesh size $h$, time step $\ddot{I}„$ and $m\ge 2$ an integer depending on the regularity of the exact solution. Finally, numerical results are provided to confirm our error estimates of the EWI-FP method and show that the convergence rate is sharp.

24 pages, 3 figures